At 6 P.M. an oil tanker traveling west in the ocean at 20 kilometers per hour passes the same spot as a luxury liner that arrived at the?

Question

At 6 P.M. an oil tanker traveling west in the ocean at 20 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 5 P.M. while traveling north at 40 kilometers per hour. If the "spot" is represented by the origin, find the location of the oil tanker and the location of the luxury liner t hours after 5 P.M. Then find the distance D between the oil tanker and the luxury liner at that time.

At what time were the ships closest together? (Hint: Minimize the distance (or the square of the distance!) between them.)

The time is ______:____ round to the nearest min

At what time were the ships closest together? (Hint: Minimize the distance (or the square of the distance!) between them.)

The time is ______:____ round to the nearest min

iceman · Answer

let t = hours after 5 PM, then 5 PM is:
t = 0 
D(t) = distance between the ships 
d₁ = oil tanker's  distance  => west
d₂  = luxury liner's distance =>north

d₁ = 20(t - 1) = 20t - 20 km west
d₂  = 40t km north

D(t) = √ [(20t - 20)^2 + (40t)^2] 
D(t) = √ (400t^2 - 800t + 400 + 1600t^2)
D(t) = √ (2000t^2 - 800t + 400)
D(t) = 20 √(5t^2 - 2t + 1)
dD/dt = [20(5t - 1)]/√(5t^2 - 2t + 1) = 0
5t - 1 = 0
t = 1/5 hr = 12 minutes 
D(1/5) = √ [(4 - 20)^2 + (8)^2] = √(256 + 64) = √320 = 8 √5 km

I hope this helps.

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At 6 P.M. an oil tanker traveling west in the ocean at 20 kilometers per hour passes the same spot as a luxury liner that arrived at the?

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